Life annuities, valuation of pension liabilities under the projected unit credit method, plan funding and the architecture of the Mexican pension system.
A pension is a life annuity: a series of payments contingent on the beneficiary remaining alive. The actuary must place a value today on an obligation that stretches decades into the future and is subject to two simultaneous sources of uncertainty: financial discounting (future money is worth less than present money) and survival (not every payment will be made). The actuarial present value combines both through the survival probability $_kp_x$ and the discount factor $v^k=(1+i)^{-k}$.
This combination gives rise to actuarial annuities, the required contributions, the reserves and, for occupational plans, the employee benefit liability that companies recognize in their financial statements under NIF D-3 (Mexico) and IAS 19 (internationally).
Life annuities are the building block of every pension. Commutation symbols allow them to be computed compactly.
Present value of $1$ payable at the start of each year to a person aged $x$ for as long as they live. It is the basis of the cost of a pension.
A person aged $x$ will receive the annuity starting in $n$ years (on reaching retirement age $r=x+n$) only if they survive. It measures the present cost of the right to retire with a pension.
Useful for pensions guaranteed for a fixed term or for valuing contributions paid only during active working life.
$D_x$ is the discounted value of the lives surviving to age $x$; $N_x$ accumulates the future $D$ values. They reduce annuities to simple ratios.
Measures the share of labor income that the pension replaces. It is the most widely used adequacy indicator for assessing a pension system.
Last survivor annuity (pays while $x$ or $y$ is alive). Essential for valuing pensions that continue to the spouse after the member's death.
The projected unit credit method (PUC, Projected Unit Credit) is the actuarial method required by NIF D-3 and IAS 19 to value defined benefits. The idea: each year of service "earns" a portion of the final benefit, projected with expected salary increases up to retirement, and that portion is brought to present value.
The projected benefit at retirement $B_N$ multiplied by the fraction of service already rendered $n_t/N$ and by the present value of the pension at retirement. It is the liability recognized on the balance sheet.
The portion of the obligation generated by one additional year of service. It is the "current service cost" charged to the income statement.
The sum of the service cost and the interest on the obligation, less, in funded plans, the expected return on plan assets $A$. Remeasurements go to OCI (other comprehensive income).
In a defined contribution plan the contribution rate $c$ (as a % of salary $S$) is set so that the present value of contributions equals that of the expected benefits.
Every pension valuation rests on assumptions: mortality table (e.g., EMSSA for active members and EMSSAH for retirees in Mexico), discount rate, salary increases, inflation, staff turnover, retirement age and, where applicable, disability and death benefits. Small changes in the discount rate move the liability disproportionately because of the long duration of the cash flows.
| Dimension | Defined Benefit (DB) | Defined Contribution (DC) |
|---|---|---|
| Promise | The amount of the pension (e.g., % of salary × years) | The contribution; the pension depends on the accumulated balance |
| Investment risk | Borne by the sponsor / institution | Borne by the worker |
| Longevity risk | Borne by the sponsor | Borne by the worker (unless a life annuity is purchased) |
| Funding | Pay-as-you-go or collective funding | Individual funded accounts |
| Balance sheet liability | Yes (DBO under PUC) | Limited to accrued contributions |
In a pay-as-you-go (PAYG) scheme, contributions from active workers pay current pensions; its sustainability depends on the demographic dependency ratio. Under funding, each generation finances its own pension through a fund that is invested. The transition from pay-as-you-go to funding creates the so-called "transition cost", since current pensions must still be paid while new contributions already flow into individual accounts.
Mexico has moved from a defined benefit, pay-as-you-go scheme to a defined contribution system with individual accounts:
The reform published on December 16, 2020 gradually increased (2023 to 2030) the employer contribution to the individual account to an average of about 15% of salary, lowered the weeks-of-contribution requirement for the PMG and redesigned the cuota social (government social contribution), aiming to raise replacement rates under the defined contribution scheme.